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Structures in Science and MetaphysicsNEWTON C. A. DA COSTA AND OTAVIO BUENO

ABSTRACT. In this paper, we develop a formalization of a meta-physics of structures and provide new applications to geometryand to metaphysics. We adopt the theory of set-theoretical struc-tures formulated in da Costa and French [15] and in da Costa andRodrigues [16]. The applications are made in the context of ex-tending the proposal advanced in Caulton and Butterfield [11] toa higher-order metaphysics.

IntroductionThis paper has three main purposes: (1) It aims to discuss some aspects of thegeneral theory of set-theoretic structures as presented in da Costa and French[15]. However, our exposition is in principle independent of the contents ofthis book, being in good part based on the paper da Costa and Rodrigues [16].The theory of structures studied here can be employed in the axiomatizationof scientific theories, particularly in physics. The theory is related to that ofBourbaki [4], but our emphasis is on the semantic dimension of the subject,which is not in agreement with Bourbaki’s syntactic views. Set-theoretic struc-tures are important, among other things, because they constitute the basis fora philosophy of science, as it was delineated in da Costa and French [15] (seealso Suppes [34]). After developing the framework, we examine some aspectsof geometry in order to make clear how the central notions of the theory canbe applied to higher-order structures found in science. The case of geometry issignificant, since pure geometric structures can be considered not only as ab-stract mathematical constructs, but also as essential tools for the formulationof physical theories, in particular the theories of space and time.

(2) The paper also aims to discuss how to extend ideas of Caulton and But-terfield [11] to what may be called higher-order metaphysics. In this meta-physics, in addition to first-order objects, we also find their correspondingproperties and relations of any type in a convenient type hierarchy.

(3) Finally, the paper outlines the first steps of the formalization of a meta-physics of structures, or structural metaphysics, inspired by ideas of authorssuch as French and Ladyman (see, for example, Ladyman [29], French andLadyman [19], and French [18]). This kind of metaphysics may be seen as anadaptation of the higher-order metaphysics of structures implicit in Caultonand Butterfield [11].

138 Newton C. A. da Costa and Otavio Bueno

1 Set theory, structures, and languagesIn this section we present the set-theoretic framework that will be appliedsubsequently in the paper, which summarizes some concepts and results of daCosta and Rodrigues [16] and da Costa and Bueno [13]. All our (set-theoretic)constructions are implemented in Zermelo-Fraenkel set theory, which is thebasis for the framework (in particular, we take languages to be a certain kindof free algebra).

The set T of types is defined as follows:

1. The symbol i belongs to T;

2. If t0, t1, . . . , tn�1 2 T, then ht0, t1, . . . , tn�1i, 1 n < w also belongs toT.

3. The elements of T are only those given by clauses 1 and 2.

The order of a type t, ord(t) is introduced as follows:

1. ord(i) = 0.

2. ord(ht0, t1, . . . , tn�1i) = max{ord(t0), ord(t1), . . . , ord(tn�1) }+ 1

The transitive closure [t] of a type t is given by the clauses:

1. t 2 [t].

2. If t = ht0, t1, . . . , tn�1i, then t0, t1, . . . , tn�1 belong to [t].

3. The elements of [t] are only those given by clauses 1 and 2.

We have: [t] = [t0] [ [t1] [ . . . [ [tn�1]. All elements of [t], different from t,have orders strictly less than the order of t. On the other hand, [i] = ∆ andord(i) = 0. There is a partial ordering in the set of types, , defined by thecondition: t1 t2 if t1 2 [t2]. Any decreasing sequence of types is finite, i.e.,the set of types is regular.

If D is a set, then we define a function tD or, to simplify, t, whose domainis T, by the following clauses:

1. t(i) = D.

2. If t0, t1, . . . , tn�1 2 T, then t(ht0, t1, . . . , tn�1i) = P(t(t0)⇥ t(t1)⇥ . . .⇥t(tn�1)), where P and ⇥ are the symbols for the power-set and theCartesian product, respectively.

The setS

range(tD) is denoted by #(D), and is called the scale based onD. The objects of t(i) are called individuals of #(D), and the objects of t(t),ord(t) > 0, are called objects or relations of type t; the type of individuals is i.We also have:

#(D) =[(range(tD)� {D}).

The cardinal k(D), defined by the condition

k(D) = sup{D,P(D),P(P(D)) . . .},

Structures in Science and Metaphysics 139

is the cardinal associated with #(D). (Sometimes, instead of k(D), we writekD.)

A sequence is a function whose domain is an ordinal number, finite or infi-nite. b

l

is the range of the sequence b, and l is an ordinal.We call structure e with basic set D an ordered pair

e = hD, ri

iwhere r

i

is a sequence of elements of #(D). We also call D the domain of e,and it is supposed to be non-empty. D and the terms of r

i

are the primitiveselements of e; the elements of D are the individuals of e. We shall identifysuch individuals with their unit sets when there is no danger of confusion.#(D) and k(D) are the scale and the cardinal associated to e, respectively. Inall cases, the ordinal which is the domain of r

i

is strictly less than k(D). #(D)is the strict scale associated to e, and contains all relations of #(D), includingthe unary relations, that is, sets.

The order of an object of #(D) is the order of its type. The order of e =hD, r

i

i, denoted by ord(e), is defined as follows: if there is the greatest orderof the objects of the range of r

i

, then ord(e) is this greatest order; otherwise,ord(e) = w.

Lw

wk(R) is the higher-order, infinitary language introduced in da Costa andRodrigues [16]. Its blocks of quantifiers are finite, and conjunctions and dis-junctions have length strictly less than k(D), when k(D) > w; if k(D) = w,such conjunctions and disjunctions are always finite. R is the set of constantsof the language, each having a fixed type.

Lw

wk(R) can be interpreted in structures of form

e = hD, ri

i,the constants denoting D and the objects r

i

, each constant possessing the sametype as that of the object it denotes. The sequence r

i

has as its domain anordinal strictly less than k(D), usually finite or denumerable. In what followswe may identify a constant with the object it denotes; this will be done whenthere is no danger of confusion. In the structure e = hD, r

i

i, D and ri

are calledthe primitive concepts of e.

Sentences of our languages are formulas without free variables. We say thata sentence x of a language, interpreted in a structure

e = hD, ri

iis true in this structure in analogy with the case of first-order (finitary) logic.To express that x is true in e we write:

e |= x.

Notions such as model, definability, etc. are defined as in da Costa and Ro-drigues [16].

From now on, we usually suppose that in the structure e = hD, ri

i the fam-ily r

i

of relations is finite (i.e., the domain of ri

is a finite ordinal). When ri

is


infinite this will be explicitly stated. As a consequence, every structure willhave, usually, an order that is a positive integer or zero.

To each structure of order n, e = hD, ri

i, it is associated a finitary languageof order n Ln

ww

(R) and an infinitary language LnwkD

(R), or to simplify Lnwk(R),

where R is the range of ri

and whose variables have types of order at mostn. These languages are the strict languages associated with the structure e oforder n.

The kind of structure e = hD, ri

i is the sequence ti

, where ti

is the type ofr

i

. The two languages associated with a given structure are both interpretablein this structure. But, for convenience, isomorphic structures of the same kindwill be identified, despite the obvious differences between them.

In the languages associated with a structure e = hD, ri

i of order n, we mayadd variables of any order m, with m > n; in particular, m may be w. Clearlysuch languages are interpretable in e = hD, r

i

i, via #(D). In this case, e be-comes the basis of a semantics of those languages. In particular, Peano arith-metic is a first-order structure, but there are corresponding arithmetic theoriesof any order whatsoever.

If e = hD, ri

i is a structure of order n, e(m) = hD, sl

i will be the structurein which s

l

is the sequence of all relations of #(D) of order m definable inthe wide sense in e = hD, r

i

i by the means of the infinitary language associ-ated with e. Sometimes, all the relations s

l

are definable in the wide sense interms of a finite number of relations in the range of s

l

. Thus, it is possibleto take s

l

as a finite sequence. e(m) constitutes the m-order counterpart of e,and the m-order semantics of e is the semantics of e(m). For example, althoughelementary Euclidean geometry is not a first-order structure, it is possible toinvestigate its first-order counterpart (see Tarski [35]).

Despite the fact that the language of set theory is first-order, structures ofany order whatsoever can be studied in it and the corresponding theory ofsets. This is, of course, a different situation from the case of elementary Eu-clidean geometry just mentioned, which involves the study of the first-ordercounterpart of a non-first-order structure. The semantic interplay between alanguage (that is, a free algebra) and a structure can be treated by means of settheory (say, Zermelo-Fraenkel set theory) even when the structure is of ordern > 1.

The theory of the kind of structure of e = hD, ri

i, or the species of struc-tures to which e belongs, can be formulated in one of its associated languages,sometimes enlarged by extra variables and constants of higher-order types, orin the language of set theory. This is accomplished by the choice of appropri-ate postulates, formulated in the chosen language.

Theories, species of structures or axiomatic systems are formulated as fol-lows: (1) It is given a language, which may be that of set theory extended bythe addition of new primitive terms, or associated with structures of a certainkind. (2) The postulates and rules of the underlying logic are those naturallyconnected with the language of clause (1) (in the case of set theory, employedas the underlying framework, its axioms are clearly included among the pos-tulates of the theory). (3) Specific postulates for the theory are also introduced.(4) By means of the logic of clause (2) and the specific axioms, the concept of


theorem is defined.In this way, one immediately obtains the common syntactic and semantic

results regarding theories. Our notion of theory, axiomatic system or speciesof structures is, therefore, more general than the one articulated in Bourbaki[4], since we are not restricted to syntactic considerations alone.

Two important concepts are those of transportable formula and of intrinsicterm that we will introduce informally. The postulates of a theory are sup-posed to be transportable formulas. A formula of the language of theory T,interpretable in structures of the kind of the structure e, is transportable if, andonly if, being true in one structure of this kind it is also true in every structureisomorphic to it.

Terms may be joined to the language of a theory as new constants or withthe help of the description operator. We say that term t is intrinsic in a theoryT if, given any two isomorphic models e and e0 of T, then for any isomorphismx between e and e0, the denotation of t in e0 is x(a), where a is the denotationof t in e.1

These considerations can be extended to the case in which the starting pointis a finite sequence D1, D2, . . . Dm of basic sets and the relations r

i

involve ele-ments of the scale #(D1 [ D2 [ . . . [ Dm). The notions of type, order of a type,scale, individuals, etc. are easily adapted to this new case. A structure, then,is a set-theoretic construct

e = hD1, D2, . . . Dm, ri

i,

where D1, D2, . . . Dm is the finite sequence of basic sets and ri

are the primitiverelations of e. The sequence D1, D2, . . . Dm (or, to abbreviate, D

d

) is composedby arbitrary (non-empty) sets, and the notions of language associated with e,theory or species of structures of the kind of e, etc. are introduced withoutdifficulty. We usually say that the sets D

d

are endowed with the species ofstructures of e.

In most cases, some or all basic sets are supposed to be endowed with pre-viously defined structures (for example, in the species of real vector spaces,the scalars are the real numbers); these basic sets are called auxiliary sets. Theother basic sets constitute the strict basic sets. Normally, structures have atleast one strict basic set.

If the family ri

in the structure e = hD, ri

i is empty, then e reduces to thebare set D.

An isomorphism between two structures of the same kind e1 = hDd

, ri

iand e2 = hB

d

, si

i is a family of bijections between Dd

and Bd

satisfying obvi-ous conditions, plus the following: the bijection between two correspondingauxiliary sets is, for simplicity, the relation of identity. In other words, iso-morphisms must keep invariant the corresponding auxiliary sets and theirassociated structures.

1The notions of transportable formula and of intrinsic term were introduced byBourbaki [4] (compare with da Costa and Rodrigues [16], and da Costa and Chuaqui[14]).


Therefore, the structure

e = hD1, D2, . . . Dm, ri

i

may be written as follows:e = hD

d

, Ad

, ri

iwhere D

d

is the family of strictly basic sets and Ad

is the family of auxiliarysets.

Structures are commonly presented as finite sequences of previously de-fined structures arranged by appropriate relations among them. So, the notionof species of structures (theory or axiomatic system) has to be adjusted to thenew situation. But that is not difficult to be done from the conceptual point ofview.

For instance, a Riemannian geometric structure (Riemannian space or Rie-mannian geometry) is an ordered pair

R = hD, mi,

where D is a (real) differential manifold and m is a Riemannian metric on D(that is, a symmetric, positively defined, 2-covariant tensor field on D). Thetheory of Riemannian spaces has as its models structures such as R = hD, mi.

Another sort of structures, usually occurring in physics, is that of a bundle.A bundle is a triplet

F = hE, p, Diwhere E (the bundle space) andD (the base space) are topological spaces, andp (the bundle projection) a continuous map of E on D.

Structures presented as sequences of other structures are reducible to struc-tures in the standard sense, although it must be clear what are the basic setsand the auxiliary sets. The reduction is expressed, among other things, by thedescription of the inter-relations imposed on the component structures. Forexample, in the case of bundles, the reduction to the standard version may beas follows:

F = hA, B, tA, tB, siwhere tA and tB are collections of subsets of A and B, respectively, and s isa function of A on B. We then introduce the postulates of F, which are thefollowing ones:

1. tA is a topology on A;

2. tB is a topology on B;

3. s is a continuous function of the topological space hA, tAi on the topo-logical space hB, tBi.

The reduction of a Riemannian structure to its standard formulation wouldbe more complex, but it can still be formulated.


2 Definability and expressibilityIn some cases, to simplify the exposition, we identify relations with theirnames. In the structure e = hD, r

i

i, we say that s 2 #(D) is strictly definablein e if the following condition is met: there is a formula F(x) of the finitarylanguage associated with e, in which x is its sole free variable, such that:

e0 |= 8x(x = s $ F(x))

in the language just mentioned and to which the new primitive term s wasadded, where e0 is the structure e with the relation s included.

Let e = hD, ri

i be a structure, #(D) its scale, s an element of #(D), andlµ

(µ < kD) a sequence of objects of e = hD, ri

i. We then define, by induction,that the expression s is expressible in the sequence l

µ

in structure e0 as follows:

1. if s is definable in the strict sense in the structure e to which we havejoined the terms of l

µ

as new primitive relations, then s is expressible inthe sequence l

µ

in e;

2. if s is not an individual of e, then s is expressible in lµ

in e if every ele-ment of s is expressible in l

µ

in e;

3. if s is expressible in the sequence ba

of elements of #(D) in the structuree and every element of b

a

is expressible in lµ

in e, then s is expressible inlµ

in the structure e.

Expressibility is equivalent to definability in the infinitary language associ-ated with e extended by the addition of the term s (see da Costa and Rodrigues[16]). Instead of ‘expressibility’, we also use the expression ‘definability in thewide sense’.

A definition of the new constant c joined to the language of the theory T isan expression of the form c = ixF(x), such that:

T ` 9!xF(x),

where F(x) is a formula of the original language of T (without c) with x as itsonly free variable.

The previous definition can be adjusted to the cases in which the constantc belongs to the language of T, or in which T has no specific postulates (suchpostulates constitute the empty set), or when the notion of semantic conse-quence (|=) is used instead of that of syntactic consequence.

Let L be a language, e1 = hD, ri

i be a structure whose language L(D, ri

) isan extension of L by the introduction of the new constant D and the familyr

i

of new primitive constants, and e2 = hD, sl

i be another structure havingits language defined analogously. In this case, e1 is said to be strictly (widely)equivalent to e2 if every s

l

is definable in the strict (wide) sense in terms of Dand r

i

, and conversely. For example, in the language of set theory, two topo-logical structures, t1 = hD, vi and t2 = hD, Ai, characterized by the family ofneighborhoods and the set of open sets, respectively, are strictly equivalent.In the same language, the structure of natural numbers, in the sense of vonNeumann, is widely equivalent to the structure of real numbers, conceived


as Cauchy sequences of rational numbers; however, there are real numbersthat are not strictly definable in terms of natural numbers, although they arewidely definable in terms of these numbers.

L is the language of set theory. We assume that the postulates of L are thecommon postulates of ZF (Zermelo-Fraenkel set theory). Suppose that T1 is atheory whose language is L(D, r

i

), where D denotes a set and ri

is a family ofrelations over D. We denote by T2 another theory whose language is (LE, s

l

),similarly defined. Under these conditions, T1 and T2 are equivalent if:

1. every sl

and E are definable by formulas of L(D, rl

) in T1, and every ri

and D are definable in L(E, sl

) by formulas of L(E, sl

) in T2;

2. every postulate of T2, reformulated in the language of T1, is provable inT1, and conversely.2

The usual metamathematical results involving mathematical structures andspecies of structures can be obtained in this approach. For instance, we have:(1) Any two models of second-order arithmetic are isomorphic. (2) First-orderarithmetic is not categorical. (3) In first-order languages, as usually formu-lated (with a first-order semantics), the notion of finite group is not axiomatiz-able. (4) In a convenient infinitary, first-order arithmetic, all subsets of naturalnumbers are widely definable. (5) In set theory, any structure of any order canbe extended to a rigid structure, by adding a finite number of new relations(da Costa and Rodrigues [16]).

3 Theories and operations with structuresAs noted, structures are correlated with, at least, three classes of language:the two classes associated with them and that of set theory. The orders of thelanguages associated with a structure e, languages that can be finitary or in-finitary, are the same as that of e. Nonetheless, as also noted, any structurecan be studied in set theory, i.e., with the help of the tools of set theory, whoselanguage is first order. We need to be careful here, since a given structureof order n has an intended semantics of order n. In a certain sense, the or-der of structure e is kept invariant in set theory, but the membership relationstrongly enriches the intended semantics of e. Concepts like definability, de-finability in the wide sense, discernibility, indiscernibility, individuality, etc.,to be examined below, are essentially language dependent.

From now on, we reserve the expression ‘species of structures’ for theo-ries axiomatized in set theory (ZF with its axioms), but whose language isextended by adding extra symbols denoting relevant sets and relations. Whena different language is employed, this will be made explicit.

Species of structures have important features. We highlight some below:

1. When a species of structures is based on various sets, it is easy to trans-form the species into an equivalent one with only one basic set. How-ever, if the underlying language of a theory differs from that of set

2There are some obvious variations in the definition of theory equivalence. But weneed not discuss them here (see, for instance, Shoenfield [33] for the case of finitary,first-order languages).


theory, for example, if it is a first-order language involving various spe-cific primitive symbols (constants), this transformation is not alwayspossible. To implement the transformation, we obviously need the re-sources of set theory. Shoenfield [33], for instance, axiomatizes whatis essentially second-order arithmetic in a two-sorted, first-order lan-guage. But it is not possible to reduce the basic sets to only one in thelanguage used. Some non-standard models are also included via thecorresponding first-order semantics.

2. The usual frameworks for the study of higher-order structures are settheory or higher-order logic. In both cases, the intended, standard se-mantics is adopted. However, a many-sorted, first-order logic can alsobe employed, with the inclusion of first-order structures.

3. The specific axioms of a species of structures are always supposed tobe transportable and its terms intrinsic. When the underlying logic of atheory is type theory (higher-order logic) these conditions are automat-ically satisfied (see da Costa and Chuaqui [14]).

4. Starting with initial structures, new structures can be obtained in settheory via various methods. The following are worth mentioning: ax-iomatization, (structural) deduction, combination, structural derivation,and the use of universal mappings (see Bourbaki [4]). We consider eachof them in turn.

Axiomatization is the general way to introduce new species of structures.This is done, as indicated above, in accordance with the definition of speciesof structures. When other languages are used, the process is similar, but withsuitable adjustments.

As an illustration, here are some examples of species of structures: (i) agroupoid is a set endowed with a binary operation; (ii) a semigroup is a groupoidwhose operation is associative; (iii) a semigroup with identity is a manoid.Moreover, species of structures such as topological space, lattice, Boolean al-gebra, loop and topological group can also be defined.

Deduction of species of structures (or structural deduction) consists in a methodof formulation of new structures that is best illustrated by an example (for adetailed description, see Bourbaki [4]). The real projective plane depends onthe species of real numbers:

R = hR,+,⇥, 0, 1,i,which is a higher-order structure. As is well known, it can be characterizedas follows: (1) The starting point is a set of triples of real numbers. (2) Projec-tive points are then introduced as classes of equivalence of triples. (3) Finally,primitive concepts of the real projective plane are introduced: the set of points(P), the set of lines (L), and the relation of incidence I (if a point and a line areincident, the point lies on the line or the line passes through the point). Thestructure of a real projection plane is, thus, a triplet:

P = hP, L, Ii


satisfying proper axioms. In other words, the real projective plane was ob-tained, by deduction, from R through the intrinsic terms P, L and I (whoseintrinsic nature can be easily verified). It is clear that the notation:

hP, L, Ii

for the real projective plane is no more than an abbreviation, omitting manydetails; nonetheless, it is a useful notation.

Numerous species of structures are obtained via combination of other species.For example, a topological group G involves the species of structures of groupand that of topological space:

G = hA, Bi,

where A is a group and B a topological space, the operations of A beingcontinuous relatively to the topology of B. Various species of structures areobtained in this way, such as differentiable manifold, analytic manifold, andGrassmann algebra.

The operation of structural derivation (of structures) is related to the notionof morphism, as discussed in Bourbaki [4]. Examples of structures of this sortare the inverse image structure, induced structure, product structure, directimage structure, and quotient structure.

Free structures are introduced with the help of universal mappings. For ex-ample, mathematically speaking, a formal language is a kind of free algebraicstructure.3

4 The Erlangen Program and some of its extensionsKlein’s Erlangen Program is well known in geometry. Veblen, for instance,summarizes it as follows:

The system of definitions and theorems which expresses prop-erties invariant under a given group of transformations may becalled, in agreement with the point of view expounded in Klein’sErlangen Program, a geometry. (Veblen and Young [36], volume I,p. 71.)

Reflecting on the concept of space, Wilder notes:

In modern mathematics, the term ‘space’ has extremely broadconnotations, the difference between ‘set’ and ‘space’ often beingvery slight, a ‘space’ being simply a set to which certain specialproperties have been added. The most common property is thathaving a metric, or distance function. (Wilder [38], p. 177.)

He then continues:

3A natural way of formulating the concepts of species of structures and of structureis to use category theory (see Corry [10]). But it would take us too far afield to examinethis issue here.


And, according to the point of view proposed by Klein in 1872,one may speak of the x-geometry of the space S as the study of theproperties of the space and its configurations that are invariantunder [the group] x. That is, x-geometry of the space S is thestudy of the x-geometry of S. (Wilder [38], p. 180.)

The concept of geometry a la Klein can, of course, be formulated in set theory(see Klein [24] and [25]). Let D be a set and g a group of transformations ofD. A geometry in the sense of Klein is a structure G = hhD, r

i

igi, such that gis the group of all transformations of D which leave relations r

i

invariant, andany invariant relation in #(D) is expressible in hD, r

i

i in terms of ri

. We call‘figure’ any element of #(D) defined by a formula of the language of G, i.e.,that of set theory plus D and r

i

. Two figures A and B are said to be equipollentif they are of the same type and there is a transformation of g that transformsA onto B. Equipollence is an equivalence relation on the set of figures of afixed type. The invariants of figures are said to be invariants of G.

In most cases, D is already endowed with some structure, and the relationsof this structure are employed to define the relations r

i

. If g is the group ofautomorphisms of e = hD, s

l

i, then e0 = hhD, sl

i, gi is a Klein geometry. It isclear, then, that the concept of geometry underlies all mathematics. (An im-portant generalization of Klein’s geometry is obtained by means of the notionof the action of a group on a structure, but here is not the place to pursue thisissue.)

Another conception of geometry is that of Blumenthal and Menger [3], towhich we now turn. Any figure F of G = hhD, r

i

igi belongs to #(D) and so canbe set-theoretically constructed starting with some subset K of D. The struc-ture hK, r⇤

i

, Fi, where r⇤i

is the restriction of ri

to K, restriction easily definable,is called the structure associated with F. Given two figures F1 and F2, we canprove the following propositions:

PROPOSITION 1. If F1 and F2 are equipollent, then their associated structures areisomorphic (as substructures of hD, r

i

i). (We show below that de converse of this firstproposition is not true.)

PROPOSITION 2. Any substructure of hD, ri

i is a figure.

Two figures are said to be BM-equipollent if their associated structuresare isomorphic. Any BM-equipollence constitutes a relation of equivalencein the class of all figures of a given type. For Blumenthal and Menger, ageometry over a structure e = hD, r

i

i is a pair H = he,⇡ti, where ⇡t is aBM-equipollence between figures of type t. Loosely speaking, in the studyof H we are interested in properties of figures that are invariant under BM-equipollences.

The connections between Klein geometries (K-geometries) and those of Blumenthal-Menger (BM-geometries) are clear: every K-geometry is a BM-geometry, but,as will become clear below, there are BM-geometries that are not K-geometries.

In the definition of BM-geometry, the structure he,⇡ti may be such that e istrivial, i.e., a bare set. Blumenthal and Menger note that:

This widens the applicability of the notion. We may, for example,


consider the theory of cardinal numbers as a geometry over a setÂ, by defining two figures (subsets) of Â to be equivalent if andonly if there exists a one-to-one correspondence between their el-ements. But in most of the important geometries, the conventionthat establishes the equivalence of figures makes use of the struc-ture by virtue of which a set becomes the element-set of a space[structure]. Hence, one speaks more often of a geometry over aspace than over a set. (Blumenthal and Menger [3], p. 28.)

Clearly, this remark is also true of K-geometry.In Riemannian geometry and some of its extensions, due to the fact that

the corresponding automorphism groups are commonly trivial, composed bythe identity transformation only, Klein’s methodology isn’t enough to specifysatisfactorily a geometry as a K-geometry. It is not difficult to realize that theinclusive view of Blumenthal and Menger cannot be profitably applied either.To address this problem, Veblen devised an interesting approach (see Veblenand Whitehead [37]).

According to Veblen, a pseudo-group is a non-empty set S of bijective trans-formations of subsets of a fixed set, such that: (1) The inverse of any transfor-mation of S belongs to S. (2) If there is the product of two transformations ofS, then it is in S. Obviously, every transformation group is a pseudo-group.

Let us now suppose, in what follows, that all functions satisfy standard con-ditions of continuity and differentiability. Veblen then defines, in a differentialmanifold M, a geometric object k as follows: (a) k is defined at every point Pof M; (b) k determines, in every coordinate system S, at P, an ordered set of mreal numbers, called the components of k in S at P (m may be different fromthe dimension of M); (c) the components of k in any coordinate system S0, at P,are functions of the components of k in S at P, the functions representing thecoordinate transformations and the derivatives of these functions evaluatedat P.

As a result, geometric objects are defined in every point of the manifold andare characterized by their transformation laws. They are, in fact, invariant un-der the pseudo-group of point transformations induced by local coordinatetransformations. Examples of these objects are: (1) scalars, i.e., objects thathave only one component, which is invariant under coordinate transforma-tions, (2) vectors, and (3) tensors.

A differential geometry, in the sense of Veblen, is a triple V = hH, g, ki, inwhich M is a differential manifold, g is a pseudo-group as above, and k is ageometric object. When k is a scalar, V is essentially the manifold M. When kis a (metric) tensor, V is a Riemannian geometry. (See Kobayashi and Nomizu[26] for a discussion of differential geometry.)

The BM-metric geometry associated to a ‘space’ with a distance functiondefined between any two of its points, following Blumenthal and Menger, isthe BM-geometry in which BM-equipollence is the relation of isomorphismbetween figures of a fixed type.

Here are some examples: (1) Geometries according to Klein: (1.1) the topol-ogy of R3 is the geometry whose group is the group the homeomorphismsof R3; (1.2) the ‘logic’ of a set is the K-geometry whose group of transforma-


tions is composed by all transformations of the set; (1.3) the affine geometry ofR3, in the present context, corresponds to the group of affinities; (1.4) the Eu-clidean metric geometry is characterized by the group of isometries of R3. (2)BM-geometries: (2.1) the metric geometry of an infinitely dimensional Hilbertspace; (2.2) the analogous metric geometry of a Minkowski space (the spaceof special relativity). (3) Veblen geometries (V-geometries): (3.1) any Rieman-nian geometry is a V-geometry; (3.2) differential manifolds are V-geometries,in which the geometric object is a scalar.

Geometries were envisaged above as set-theoretic structures. However,they can also be considered as species of structures or, more in line withcurrent mathematical practice, as certain kind of theories. The axiomatic ap-proach to geometry is firmly based on language, as is evident. In variouscases, this language is that of set theory extended by new primitive terms (andpostulates), but typed languages can also be employed, for instance those as-sociated with the structures under study. Even when a particular geometryis formulated in a higher-order framework, it is possible, as we have alreadyobserved, to investigate its lower parts. This happens in the cases of the Eu-clidean real plane and of arithmetic (another example: we could investigatefirst-order versions of special relativity). (Axiomatic geometry, however, willnot be taken into consideration here, since it is not relevant to the objectives ofthis paper.)

Let us return to structures, and let t = hE, Ti be a topological space insideZF, in which E is the base space and T its topology. The language in which wetalk about t is, thus, that of ZF plus the constants t, E and T; the postulates arethose of ZF plus the common topological axioms with some extra ones (suchas those of Hausdorff). We denote by G the group of homeomorphisms (theautomorphisms) of t. Clearly any topological notion, that is, an element of thescale of t, definable in the structure in the strict sense or in the wide sense, isinvariant under G. But an important point is also the converse question: Isany invariant notion definable in function of the primitive notions of t? In thisconnection, we have:

THEOREM 3. In t, any invariant notion d (that is, an element of the scale of t) isdefinable in the wide sense in the infinitary language associated with t whenever d isinvariant under G.Proof. To say that d is definable in the wide sense in t means that d is expressible inthe primitive constants of t. So, the theorem is a consequence of the results of da Costaand Rodrigues [16].THEOREM 4. In Euclidean metric geometry, a notion is definable in the wide sense(that is, expressible) if, and only if, it is invariant under the group of isometries.Proof. See da Costa and Rodrigues [16].

In general, propositions analogous to the preceding ones are valid for allstructures.

THEOREM 5. There exist BM-geometries that aren’t K-geometries.Proof. It suffices to observe that the BM-metric geometry of an infinite-dimensionalHilbert space is the study of isometries between figures of the space, and that there areisomorphisms between two figures that cannot be extended to an isometry of the whole


space onto itself.

THEOREM 6. There are structures in which some notions are expressible (definablein the wide sense) but not definable in the strict sense in the denumerable languageassociated with the structure.Proof. In ZF the set of real numbers R is rigid: R has only one automorphism, itsidentity isomorphism. As a result, according to da Costa and Rodrigues [16], anynotion of the scale of R is expressible, i.e., definable in the wide sense, in the infinitarylanguage associated with R. However, since R is not denumerable, there are realnumbers that are not definable in the (finitary) language of ZF plus the primitivesymbols corresponding to primitive notions of R.

We also have:

THEOREM 7. Any structure can be extended to a rigid structure by the addition ofa finite number of new primitive relations.Proof. The proof of this theorem of Sebastiao e Silva is presented in da Costa andRodrigues [16].

The Galois group of a Riemannian geometry V is usually trivial. So, in thiscase, any notion (element of the scale of V) is only definable in the wide sensein V. Moreover, there are notions that are not definable in the strict sense inV, with the help of the underlying language of ZF, as is clear.

5 Indiscernibility and related conceptsSince the beginning of quantum theory, in the early part of the twentieth cen-tury, notions such as indiscernibility and identity have been repeatedly dis-cussed in the context of physics. For instance, Schrodinger argued, severaltimes, that the notion of identity is meaningless in connection with elemen-tary particles (see French and Krause [20] for a thorough discussion of thisissue).

We now consider the analysis of the notions of indiscernibility, identity, andindividuality in the domain of geometry. We take this to be a first step towarda future analysis of these concepts in the context of quantum physics. A keymotivation here is the work done by Caulton and Butterfield [11].

Recall that our discussion is carried out in set theory, in which languagesare both set-theoretic constructs and geometric structures. Geometry has, ofcourse, two main faces, since there is a parallelism between geometry as amathematical discipline and geometry as a physical theory underlying otherphysical theories. Both can be formulated in set-theoretically.

Identity, informally, constitutes the connection between objects a and b whichare the same. We write, in this case, a = b. In set theory, ‘identity’ has twomeanings: it may denote a basic concept of set theory or it may refer to the di-agonal of the Cartesian product of a non-empty set by itself, i.e., a set-theoreticrelation, formed by a set of ordered pairs whose two coordinates are the same.It is clear that, in set theory, infinitely many relations of identity can be ex-pressed. In what follows, the reader will easily recognize the meaning inwhich term ‘identity’ is being employed.

Our set-theoretic ZF system is a pure set theory, in the sense that the only


objects taken into consideration are sets. Identity, in the two set-theoreticsenses above, can be defined or can be taken as a primitive notion. (On whetheridentity in general — understood as a fundamental concept rather than as anotion in set theory — can in fact be defined is an issue explored in Bueno [8]and [9].)

Geometric objects belong to scales of structures. So identity commonlyrefers to objects in certain geometric structures that are associated with lan-guages. To talk about identity, in this case, is normally to talk about relationsin the scales of structures, associated with abstract languages.

Identity is independent of language: if a and b are, for instance, figures inthe scale of structure S, then that a is identical to b (or is different from b)constitutes a relation that is independent of the language associated with S.But this is not true of the notion of indiscernibility, which generalizes the con-cept of identity. We distinguish three kinds of indiscernibility: the weak, thestrong, and the intrinsic kinds. They extended concepts presented in Caultonand Butterfield [11], although we use a different terminology.

5.1 Weak indiscernibilityBy e = hD, r

i

i we denote, in the next definitions, a first-order structure inwhich r

i

is a finite sequence, and Le is the first-order finitary language asso-ciated with e, and x and y are two distinct variables of Le. We then have thefollowing definitions, in which r stands for any element of the sequence r

i

.DEFINITION 8.

1. If r is a unary predicate the formula:

r(x) $ r(y)

is called the companion formula of r.

2. If r is a binary relation, then the formula:

8z[(r(x, z) $ r(y, z)) ^ (r(z, x) $ r(z, y))]

is the companion formula of r.

3. If r is a ternary relation, then

8z8t[(r(x, t, z) $ r(y, t, z))^ (r(t, x, z $ r(t, y, z))^ (r(t, z, x) $ r(t, z, y))]

is the companion formula of r.

4. The same goes for any n-ary relation r, with n finite and greater thanthree.

Note that new variables t, z, . . . are different from x and y, and any two ofthem are distinct.DEFINITION 9. x ⌘ y abbreviates the conjunction of all companions of theelements of r

i

.DEFINITION 10. If a and b are numbers of the domain of e = hD, r

i

i, we saythat they are weakly indiscernible (wi) if a ⌘ b is true in e.


Definitions 1, 2 and 3 can be modified to accommodate first-order lan-guages and structures with infinitely many primitive relations, as well as in-finitary languages. We now consider higher-order languages and structures.4

DEFINITION 11. If e = hD, ri

i is a higher-order structure with one of its as-sociated languages, and a and b are elements of the scale of e, then:

1. If a and b have the same type and if there are companions of all or someof the relations r

i

, then a ⌘ b is similarly defined as in the case of first-order languages and structures. Under these assumptions, a is weaklyindiscernible from b if, and only if, a ⌘ b is true in #(D); otherwise, x ⌘y is x = x ^ y = y and a and b are also said to be weakly indiscernible.

2. If a and b aren’t weakly indiscernible, then a and b are weakly discerni-ble.

Here are some examples: (a) R3, with the standard metric d, constitutes arigid structure whose Galois group is composed by the identity transforma-tion. The same goes for the Klein geometry K = hhR3, di, gi, in which g isits group. As a result, in the associated infinitary languages, any two distinctpoints of R or of K are weakly discernible. The same point applies to twodistinct spheres. (b) Any two points of an abstract projective plane are indis-cernible (see, for instance, Heyting [22]).

5.2 Strong (or syntactic) indiscernibilityLet a and b be two figures of the same type (in particular, points) of the geometrya la Klein e = hhD, r

i

i, gi. In this case, a and b are strongly discernible if thereis a formula F(x), of the language associated with e, which has x as its solefree variable, such that, in #(D), it is true that:

|= F(a) and |= ¬F(b).

If there is no such formula, a and b are said to be strongly indiscernible. It isclear that this definition can be extended to any structure, and depends on thelanguage associated with it.

As for examples, we have: (1) In the usual higher-order, metric geometryof R3, two spheres, whose centers are strongly indiscernible, are also stronglyindiscernible. (The language of the usual geometry is finitary and, as a con-sequence, there are real numbers that are strongly indiscernible.) (2) In theusual axiomatizations of Euclidean geometry, for example the one carried outby Hilbert, any two points are always strongly indiscernible.

5.3 Essential (or semantic) indiscernibilityIn the Klein geometry G = hhD, r

i

i, gi, any two figures a and b are said tobe essentially indiscernible if there is a transformation of g that maps a onto

4On higher-order model theory and infinitary model theory, which have connec-tions with the general theory of structures, see, for example, Manin [30] and the refer-ences therein. Shelah insists that, in higher-order model theory, the central feature isnot the language but certain algebraic properties of families of structures. In particular,it is not possible to study properly such structures in first-order languages.


b. Obviously, this definition can be extended to any structures whatsoever.When two figures of the scale of a structure are not essentially indiscernible,they are essentially discernible.

Here are a couple of examples: (1) In the metric geometry of R3, figuresof the same type with domains essentially indiscernible are essentially indis-cernible. (2) In the usual propositional calculus, considered as a free Booleanalgebra, any two generators are essentially indiscernible.

Two other kinds of indiscernibility (or of discernibility), investigated byCaulton and Butterfield [11], can be extended to higher-order languages or toinfinitary ones. However, we will not examine them here. We only note thatCaulton and Butterfield’s absolute discernibility corresponds, in first-orderlanguages, to our syntactic discernibility.

There are connections among our three concepts of discernibility (and in-discernibility) that hold for all mathematical structures. Using ideas of daCosta and Rodrigues [16], the following theorems can be proved:

THEOREM 12. In any structure e = hD, ri

i, with the language Le, if two objects ofe are syntactically discernible, then they are semantically discernible.THEOREM 13. The converse of theorem 5.1 is not true in general.THEOREM 14. Under the conditions of Theorem 5.1, if Le is the strict infinitarylanguage associated with e (see Section 1 above), then syntactic discernibility is equiv-alent to semantic discernibility.

Let e = hD, ri

i be a structure and Le its language. An individual (or anindividual of type i) is an element of D that is syntactically indiscernible fromall other elements of D. More generally, an individual of type t is an elementof type t that is syntactically indiscernible from all other elements of type t.

In some cases, the objects of type t, t 6= i, can be the basic elements of thedomain of a new structure and, in this new structure, they are individuals(of type i of the new structure). It is clear that the notion of indiscernibilitydepends not only on the structure we are considering, but also on the languageemployed.

In first-order languages, weak indiscernibility can be used as a kind of iden-tity (see, for instance, Caulton and Butterfield [11]), although clearly it isn’tidentity per se. The situation in higher-order structures and languages is en-tirely different. In this case, weak indiscernibility is too restrictive to havea similar role. In effect, the primitive objects of #(D) that correspond to thestructure e = hD, r

i

i don’t have the same order and type, and ri

belongs tovarious types, so that weak indiscernibility becomes vacuous. Perhaps wecould introduce weak indiscernibility for each type. However, it is better touse identity. But in the case of first-order structures or first-order languages,weak indiscernibility is relevant.

Most of the discussion in Caulton and Butterfield [11] that is devoted tofirst-order structures and languages can be extended to the higher-order case.For instance, in elementary Euclidean geometry, that is, Hilbert geometry, twospheres with the same radius are syntactically indiscernible. However, in themetric geometry of R3, any two spheres with radii of identical length are syn-tactically discernible when their centers are also syntactically discernible.


With regard to Caulton and Butterfield’s metaphysical views, their discus-sion can also be extended to the higher-order situation. To fix our ideas, wemention that their considerations on discernibility of pairs of objects are im-mediately generalizable to higher-order objects. In fact, several of their meta-physical views are adaptable to higher-order situations. In other words, theirfirst-order metaphysics is extendable to a higher-order one. As an illustration,there can be higher-order individuals, similarly to the first-order individualsdefined by them.

6 Structural metaphysicsIn this section we outline a metaphysics of structures. Our preliminary anal-ysis is intuitive and informal, but we will indicate how the approach can bedeveloped rigorously and formally. Our work is not exegetical, and we arenot attempting to systematize the views defended by contemporary structuralmetaphysicians (see, for instance, Ladyman [29], French and Ladyman [19],and French [18]). Our aim is to present a possible metaphysics of structuresinspired by these views.

Consider the notion of mathematical structure:

e = hD, ri

ias introduced above. A particular case of structure is, obviously, that in whichthe sequence of primitive relations is empty:

e = hD, ∆i.When D = ∆, we have the empty structure that was not discussed above.When r

i

is ∆, the structure e can be considered as the set D. So, from aninformal stance, it seems reasonable to regard sets as a kind of structure.

Moreover, any relation is a strict relation, that is, an n-ary relation withn � 2, or a unary relation or set. Furthermore, relations are sets or, what is thesame thing, unary relations. In particular, a strict n-ary relation, n � 2, is alsoa unary relation, that is, a set.

The structure:e = hD, r

i

iis, basically, the binary relation:

{hD, ri

i}.

Conversely, an n-adic relation r, n � 2, essentially constitutes the structure:

r⇤ = hK, ri,where K is a set

S{{x} : x 2 r}.As a result of these informal considerations, sets, relations and set-theoretic

structures are, as it were, different faces of the same kind of basic objects:metaphysical structures or, to simplify, M-structures. Certain M-structures arefundamental: they are the M-sets, the bricks that compose all M-structures.


We can postulate, then, that M-sets constitute a system satisfying the ax-ioms of a traditional set theory, such as ZFC, Zermelo-Fraenkel set theorywith the axiom of choice (see Fraenkel and Bar Hillel [17], and Kuratowskiand Mostowski [28]). However, the postulates of the theory should not in-clude the axiom of regularity, given our assumption that any M-structure canbe formed by other structures. In this way, our system reduces to ZF.

In addition to these considerations, it is natural to take ZF as providing asystematization of a metaphysics of M-structures, independently of its stan-dard interpretation as a theory of usual sets or collections. However, we needto accommodate M-structures that are invoked in the representation of thephysical universe (or, in a metaphysical reading, in the constitution of thatuniverse). For this reason, we need to accommodate space and time. Accord-ingly, we add to ZF a new unary predicate �, according to which �(x) meansthat x is a physical (or concrete) M-structure or M-Urelement (for example,particles may be conceived of as unary M-relations, or M-structures, etc.) Theresulting system will be denoted by ZF�. Postulates governing M-structuresthat satisfy � are added to those of ZF, depending on the view one has on thenature of the universe. We are also free to strengthen ZF� in various ways,for instance in order to handle M-categories and M-toposes interpreted as M-structures. (On ZF with Urelemente, see Brignole and da Costa [5].)

We can now formulate standard scientific theories in an ontology of M-structures in analogy with the use of common set-theoretic structures in thefoundations of science. In particular, fields and particles can be interpretedin terms of different kinds of M-structures. The philosophical simplicity thatan ontology of M-structures brings to the philosophy of science, especially ofphysics, should be apparent.

Note that although ZF� was formulated in analogy with set-theoretic ideas,it does possess other interpretations, even non set-theoretic ones. This is thecase of the pure M-structures, which can be used to capture some features ofstructuralist approaches in the philosophy of mathematics and science.

M-structures can also be used to formulate Hughes’ suggestive idea ofstructural explanation. As he notes:

The idea of a structural explanation can usefully be approachedvia an example of special relativity. Suppose we were asked toexplain why one particular velocity (in fact the speed of light) isinvariant across the set of inertial frames. The answer offered inthe last decade of the nineteenth century was to say that measur-ing rods shrank at high speeds in such way that a measurement ofthis velocity in a moving frame always gave the same value as onein a stationary frame. This causal explanation is now seen as seri-ously misleading; a much better answer would involve sketch-ing the models of space-time which special relativity providesand showing that in these models, for a certain family of pairsof events, not only is their spatial separation x proportional totheir temporal separation t, but that the quantity x/t is invariantacross admissible (that is, inertial) coordinate systems; further, forall such pairs, x/t always has the same value. This answer makes


no appeal to causality; rather, it points out structural features ofthe models that special relativity provides. It is, in fact, a struc-tural explanation. (Hughes [23], pp. 256-257.)

M-structures, suitably developed, provide a straightforward way of for-mulating the relevant models of special relativity and the invariances acrossvarious coordinate systems.

One need not, however, reify these invariant features of the models: theyindicate traits that, according to the models under consideration, have a signif-icant form of objectivity. It is not up to us that these invariant features emerge:they are stable properties of the relevant models. Of course, realists will readthese traits as salient features of the world, whereas anti-realists will resistsuch interpretation. What matters to us is the understanding that emergesfrom each interpretation and the fact that both can be captured in terms ofM-structures. Explanation, including structural explanation, is clearly tied tounderstanding, and M-structures provide a rich framework to explore all ofthese alternatives. (We return to this point below.)

Realist views of structures insist that structures either provide us with waysof knowing the world (epistemic conceptions) or articulate ways the world is(ontic views). (For the distinction between epistemic and ontic views of struc-tural realism, see Ladyman [29]; see also French [18] for additional discussion.For a critique of ontological conceptions of structure, see Arenhart and Bueno[1].) In contrast, anti-realist views of structure resist such commitments, andinsist that structures, despite their help in systematizing information about theworld, should not be reified. Due to its partiality, incompleteness, imprecisionor inaccuracy, the information encoded in such structures is at best partial (fora thorough defense of the significance of partiality in this context, see da Costaand French [15]; see also Bueno [6]).

The proposal here is best thought of as being neutral between either sideof the debate: it need not be committed to either realism about structures orto anti-realism about them. Structures can be used in geometry, physics ormetaphysics quite independently of their particular interpretation, quite in-dependently of the particular framework in which they are formulated. Settheory can be employed to formulate a K-geometry without stating that thisgeometry metaphysically is a particular set-theoretic structure. One can sim-ply use set-theoretic resources as representational tools rather than as constitutivedevices. That is, set theory provides a framework to represent and formulatevarious structures, but the framework itself need not be taken to answer themetaphysical question of the nature of the objects and structures that are rep-resented in it. Nor is the framework indispensable. One can, after all, use dif-ferent frameworks to carry out the task at hand: for instance, in addition to themany non-equivalent formulations of set theory, category theory, type theory,second-order mereology, among other frameworks, can all be employed aswell. And in each of these frameworks, importantly different answers aregiven to the metaphysical question of the nature of the relevant structures:categorical, type-theoretic, mereological, etc. Since the metaphysical nature ofthe objects and structures is different in different frameworks, and all of theseframeworks are possible, it’s unclear that any particular metaphysical answer


is ultimately settled.Even M-structures, despite being decidedly formulated in set theory, need

not be taken to be metaphysically set-theoretic objects, since there are corre-sponding counterparts to such structures in all of the different frameworksmentioned above. On this interpretation, set theory is used as a representa-tional tool only without any indication of the fundamental nature of the struc-tures that are thus represented.

This distinction between set theory as a tool of representation and as a con-stitutive device can be used in other contexts in which set theory is employed.As is well known, Tarski often highlighted that he was a nominalist (for refer-ences and discussion, see Frost-Arnold [21]). Given all the work he did in settheory as well as his use of it in semantics and theories of truth, if set theorywere ontologically committed to sets as abstract objects (as the usual platonistreading would have it), Tarski would end up being incoherent. However, ifset theory is simply interpreted as a representational tool, which need not cap-ture or state the fundamental nature of the objects under study, one can useit while still resisting commitment to the existence of the objects and struc-tures it refers to. (In addition, neutral quantifiers, which do not presupposethe existence of the objects that are quantified over, can also be invoked. Thusquantifying over sets is not enough to be ontologically committed to their ex-istence: additional conditions of an existence predicate also need to be met;for details, see Azzouni [2] and Bueno [7].) This has the advantage of makingTarski’s stance coherent, while providing an alternative setting to understandthe use of structures in a variety of domains, from geometry through physicsto metaphysics.

We noted above the significance of structural explanations (and theoreticalexplanations) in physics. Are there such explanations in metaphysics? Clearlythere should be. Certainly there are no causal explanations in this area, but stillexplanatory devices are systematically invoked. A metaphysical problem canbe thought of as something that typically emerges from incompatible situa-tions in (our conceptual description of) the world. Metaphysical explanationsare then attempts to resolve the (apparent) tension. These explanations areconsiderations that indicate the possibility of certain situations in light of cir-cumstances that seem to undermine them. For instance, how are individualspossible at the fundamental level given some interpretations of non-relativistquantum mechanics that deny that identity can be applied to quantum par-ticles? How can abstract (that is, causally inert, non-spatial-temporal) mathe-matical structures exist given a world constituted by concrete (causally active,spatial-temporal) entities? Metaphysical explanations indicate that the appar-ent tension between these circumstances can be resolved. In some cases theysuggest ways of reconciling the conflicting situations; in others, they providereasons to undermine some of the built-in assumptions invoked in the de-scription of the circumstances under consideration; in yet others, they suggestways in which the inconsistency identified among the relevant situations canbe tolerated.

In each case, some understanding emerges: we may realize the consistencyof situations we thought were actually impossible; we may identify assump-


tions for which we may lack good reason to accept; we may learn about abroader range of possibilities (which may eventually include what was pre-sumed to be, up to that point, impossibilities). In each case, our understandingincreases, by gaining information about ways of reconciling what was thoughtto be incompatible, by obtaining information regarding false or questionableassumptions, or by increasing the scope of what is deemed possible. Meta-physical explanations are, thus, crucial for our understanding of several as-pects of our conceptual framework. (For the general structure of philosophi-cal explanations as ways of dispelling apparent tensions between conflictingassumptions, see Nozick [31].)

We think that M-structures provide a rich framework in which we canexamine and assess these metaphysical explanations as they emerge in thefoundations of the sciences. We can clearly identify the relevant assumptions:set-theoretic, category-theoretic, type-theoretic, mereological, etc., dependingon the context, as well as empirical, theoretical and conceptual presupposi-tions. We also probe each framework’s expressive and inferential power andits limitations. As a result, a proper assessment of their pros and cons canbe determined and, in this way, some progress is eventually made. And ourunderstanding, in the end, increases.

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